The mage was complicated; the archmage, even more so. Strap in.
As with the mage, the archmage’s ability scores imply an aversion to melee combat, a strong self-preservation impulse, and a strategically and tactically savvy view of the battlefield. We can also determine, by reading between the lines, that the archmage is a wizard of the abjuration school, because Magic Resistance is a feature that abjuration wizards obtain at level 14. Mechanically, that doesn’t mean much, since Magic Resistance is the only abjuration feature the Monster Manual gives the archmage; still, this inference adds a dash of flavor to our archmage’s personality. Abjuration is the magic of prevention. All other things being equal, the archmage’s primary impulse is to shut you down.
The mage’s spells topped out at 5th level, but the archmage’s go all the way up to 9th, with only one slot each of the top four levels. I assume this was to simplify an already extremely complicated and powerful enemy. What it means for us is that those four slots are reserved exclusively for the archmage’s four highest-level spells: time stop, mind blank, teleport and globe of invulnerability. And mind blank, according to the MM, is pre-cast before combat begins—along with mage armor and stoneskin—so that slot isn’t even available. (How does the archmage know to combat is about to begin? Dude, 20 Intelligence. “Nolwenn an Gwrach, Supra-genius!”)